Stat 567:Statistical Reliability
Fourteenth Class (October 11, 1996)


Maximizing Likelihood Functions
Using JMP's Nonlinear Fit Platform

Class Objectives:

Homework Assignments:

Class Outline and Main Points:

Maximizing Likelihood Functions Using JMP's Nonlinear Fit Platform

We illustrate how JMP calculates maximum likelihood estimates using the Ball Bearing data of Example 2.1, Page 37 of your textbook. In particular, we obtain the MLEs for the parameters of a Gumbel model applied to the log data. From these MLE values we can obtain the MLEs for the parameters of the Weibull model of the original data. This follows since the Weibull scale parameter is log(mu) and the Weibull shape parameter is 1/sigma where mu and sigma are the location and scale parameters of the Gumbel model of the log data.

Here are the steps one must go through with JMP:

Notice that JMP arrived at the maximum likelihood estimates given in the book. It also calculated the equivalent of the variance-covariance matrix, namely, the standard errors and the correlation of the estimates. In addition, it calculated the likelihood ratio confidence intervals. As we have discussed these are usually better than the confidence intervals based on standard errors.

Verifying Optimization Results Obtained Using JMP's Nonlinear Fit Platform

The results of an iterative numerical optimization technique should be verified. One could have converged to a local minimum or an inflection point rather than to the global minimum.

We illustrate how JMP can be used to verify the MLEs obtained using the Nonlinear Fit platform. We continue with the ball bearing example at the point of the last JMP screen shot shown above.

Here are the steps one must go through with JMP:

mu = 4.40
sigma = 0.4758

(Notice that we have only checked a neighborhood around the MLE values computed by JMP. Hence, we cannot be completely sure that these values are really the one that minimize the negative of the log-likelihood function. Nevertheless, we are very happy with the results of the contour plot!)

A fancier - though less useful - way to show that JMP's Nonlinear platform arrived at the true minimum of the negative log-likelihood function is to use JMP's Spinning Plot:

Below are the steps one must go through with JMP:

This spinning plot is very impressive when shown in a computer screen or projector.


Maximizing Likelihood Functions Using JMP's Nonlinear Fit Platform When the Data is Right Censored.

We use the data on the strengths of 48 pieces of weathered braided cord (Example 2.3, Page 46 of your textbook ) to illustrate the steps that one needs to go through to obtain MLEs in the right censored case.

The following screen shots should make it clear how to modify the steps provided above to accommodate right censoring.








As one can see here we also obtain the MLEs for the Weibull model of the cord data. Notice that the values coincide with those provided in the textbook.

Study Questions

SAS JMP files (Mac) of classroom examples and homework

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